Local description of band rearrangements. Comparison of semi-quantum and full quantum approach

TL;DR

This paper compares semi-quantum and full quantum approaches to understanding band rearrangements in molecules, demonstrating how topological invariants relate to quantum level transfers near degeneracy points.

Contribution

It introduces a local semi-quantum Hamiltonian framework and links it to a Dirac operator model for full quantum analysis, revealing topological effects on energy level crossings.

Findings

Delta-Chern invariants characterize band rearrangements.

Quantum level transfer corresponds to boundary crossing with invariant ±1.

Full quantum model uses Dirac operator with specific boundary conditions.

Abstract

Rearrangement of rotation-vibration energy bands in isolated molecules within semi-quantum approach is characterized by delta-Chern invariants associated to a local semi-quantum Hamiltonian valid in a small neighborhood of a degeneracy point for the initial semi-quantum Hamiltonian and also valid in a small neighborhood of a critical point corresponding to the crossing of the boundary between iso-Chern domains in the control parameter space. For a full quantum model, a locally approximated Hamiltonian is assumed to take the form of a Dirac operator together with a specific boundary condition. It is demonstrated that the crossing of the boundary along a path with a delta-Chern invariant equal to $\pm1$ corresponds to the transfer of one quantum level from a subspaces of quantum states to the other subspace associated with respective positive and negative energy eigenvalues of the local Dirac Hamiltonian.